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arXiv:2605.17974

Measuring the complete set of spatial Schmidt modes of entangled two-photon fields

Radhika Prasad, Nilakshi Senapati, Abhinandan Bhattacharjee, Suman Karan, Anand K. Jha

quant-phSubmitted May 18, 2026• Announced May 19, 2026• 0 citations
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Abstract

Spontaneous parametric down-conversion (SPDC) is the most widely-used source of high-dimensional entangled two-photon states, and the entanglement in the spatial degree of freedom is considered best suited for harnessing high-dimensional advantages. Although the Schmidt basis provides a natural choice for state characterisation of entangled two-photon states in any degree of freedom, there is currently no technique that can measure the Schmidt basis of an entangled two-photon field. The existing techniques can only reconstruct the Schmidt spectrum when the Schmidt basis is known a priori. In contrast, we present a technique that measures the complete set of spatial Schmidt modes without any prior knowledge. Using this technique, we report measurement of states with over 3000 Schmidt modes -- highest reported yet -- with up to 98$\%$ fidelity. We expect our work to significantly advance the harnessing of high-dimensional advantages in SPDC-based systems.

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ACS Photonics•May 28, 2025•5 citations
Spontaneous parametric down-conversion (SPDC) is the most widely-used source of high-dimensional entangled two-photon states, and the entanglement in the spatial degree of freedom is considered best suited for harnessing high-dimensional advantages. Although the Schmidt basis provides a natural choice for state characterisation of entangled two-photon states in any degree of freedom, there is currently no technique that can measure the Schmidt basis of an entangled two-photon field. The existing techniques can only reconstruct the Schmidt spectrum when the Schmidt basis is known a priori. In contrast, we present a technique that measures the complete set of spatial Schmidt modes without any prior knowledge. Using this technique, we report measurement of states with over 3000 Schmidt modes -- highest reported yet -- with up to 98%\%% fidelity. We expect our work to significantly advance the harnessing of high-dimensional advantages in SPDC-based systems.
High-dimensional spatially entangled two-photon state generated by spontaneous parametric down-conversion process (SPDC) has become a promising resource for several quantum information science applications. For harnessing high-dimensional entanglement advantages, detection capability in the Schmidt basis is a necessity. Spatial entanglement has been explored in several modal bases, such as pixel, azimuthal, and radial modes. Among them, pixel and azimuthal entanglement have been widely utilized due to efficient access to their Schmidt modes, while radial-mode entanglement remains underexploited. This is because for radial coordinates, there is neither a Schmidt-decomposed form for the SPDC photons nor is there a technique for measuring high-dimensional radial Schmidt modes, which is a major roadblock in harnessing radial mode advantages. In this work, we first theoretically show that the azimuthal averaging of SPDC two-photon state yields a radial Schmidt-decomposed form under typical experimental situations. We then demonstrate an innovative approach for extracting the radial Schmidt modes and their spectrum by characterizing the density matrix in the radial basis of one of the SPDC photons. Finally, we report the first-ever measurement of radial Schmidt spectrum of upto 50 radial Schmidt modes with about 98\% fidelity.
The full Schmidt decomposition of spatiotemporally entangled states generated from spontaneous parametric down-conversion (SPDC) has not been carried out until now due to the immense computational complexity arising from the large dimensionalities of the states. In this Letter, we utilize the rotational symmetry of the states to reduce the complexity by at least four orders of magnitude and carry out the decomposition to reveal the precise forms of the spatiotemporal Schmidt modes and the Schmidt spectrum spanning over 10^4 modes. We show that the Schmidt modes have a phase profile with a transverse spatial vortex structure that endows them with orbital angular momentum at all frequencies. In the high-gain regime, these Schmidt modes broaden and the Schmidt spectrum narrows with increasing pump strength. Our work can spur novel applications at the intersection of quantum imaging and spectroscopy that utilize entangled states produced from SPDC.
Abstract High-dimensional spatially entangled two-photon states generated by the spontaneous parametric down-conversion process (SPDC) have become a promising resource for several quantum information science applications. In order to harness high-dimensional entanglement advantages, detection capability in the Schmidt basis is a necessity. Spatial entanglement has been explored in several modal bases, such as pixel, azimuthal, and radial modes. Among them, pixel and azimuthal entanglement have been widely utilized due to efficient access to their Schmidt modes, while radial-mode entanglement remains underexploited. This is due to radial coordinates not having a Schmidt-decomposed form for the SPDC photons or a technique for measuring high-dimensional radial Schmidt modes, which is a major roadblock in harnessing radial mode advantages. In this work, we first theoretically show that azimuthal averaging of SPDC two-photon states yields a radial Schmidt-decomposed form under typical experimental situations. We then demonstrate an innovative approach for extracting the radial Schmidt modes and their spectrum by characterizing the density matrix in the radial basis of one of the SPDC photons. Finally, we report the first-ever measurement of the radial Schmidt spectrum of up to 50 radial Schmidt modes with about 98% fidelity.
High-dimensional entangled states have now been identified as excellent candidates towards the enhancement of the bandwidth of quantum systems. A variety of methods exist that aim to certify and set the bounds to entanglement. However, rapid and accurate approaches for precise quantification of the dimensions remain a challenge. Here, we report a facile, rapid and robust approach that quantifies a wide range of the spatial dimensions of high-dimensional entangled states using an interferometric technique using only two images. Our process works for a class of pure two-photon states, regardless of its separability and provides a fast and easy way to accurately measure the spatial Schmidt number which is a quantitative measure of dimensionality of entanglement. We apply our method to two-photon states generated by spontaneous parametric down-conversion (SPDC) and show that our results are in excellent agreement with numerical estimates. Since such estimates only exist for collinear phase matching in SPDC crystals, we also derive a noncollinear phase-matching condition applicable for type-2 crystals. In these nonlinear crystals, the commonly applied assumption of degenerate daughter photons no longer holds, leading to deviations from earlier theoretical predictions. Our interferometric technique provides excellent results for both type-0 and type-2 collinear phase-matching conditions, endorsing its wide applicability in quantum photonics. Published by the American Physical Society 2024
High-dimensional photonic entangled states are key resources for advancing quantum imaging, quantum sensing, secure communication, and quantum computation. The most common method for generating such states is spontaneous parametric down-conversion (SPDC), in which a pump photon splits into two daughter photons that share correlations across multiple photonic degrees of freedom. Characterising these high-dimensional states, however, is highly challenging, as it scales quadratically with the state dimensionality and usually requires large sets of projective measurements within the framework of positive operator-valued measures (POVMs).
Entanglement is a unique feature of quantum mechanics. In coupled systems of light and matter, entanglement manifests itself in the linear superposition of multipartite quantum states (e.g., parametrized by multiple spatial, spectral, or temporal degrees of freedom of a light field). In bipartite systems, the Schmidt decomposition provides a modal decomposition of the entanglement structure over independent, separable states. Although ubiquitous as a mathematical tool to describe and measure entanglement, there exists no general efficient experimental method to decompose a bipartite quantum state onto its Schmidt modes. Here, we propose a method that relies on bipartite self-configuring optics that automatically “learns” the Schmidt decomposition of an arbitrary pure quantum state. Our method is designed for entanglement distributed over spatial or spectral degrees of freedom of a light field and can reconstruct the Schmidt modes and values by variational optimization of the network’s output powers or coincidences. We illustrate our method with numerical examples of spectral entanglement analysis for biphotons generated via spontaneous parametric down conversion and provide experimental guidelines for its realization and generalization to other degrees of freedom of light, including the influence of losses and impurities. Our method provides a versatile and scalable way of analyzing entanglement in bipartite integrated quantum photonic systems.